I usually use my cameras in Av, Aperture Priority, mode, regardless of whether I am prioritising aperture or shutter speed. As the two are interlinked to obtain correct exposure, if I want a fast shutter speed I just open up the aperture until I have the value I want. If this compromises depth of field too much, then I also work at a different ISO or use a tripod. So, no need to keep changing mode. Of course, it's equally viable to work the other way round, and some photographers may well prefer to use Tv, or Shutter Speed Priority.
But whatever method of getting there, slow shutter speeds will need a tripod or Shake Reduction/Steady Shot/Image Stabilisation/Vibration Reduction and this does change the game somewhat. However, for still subjects, a camera on a firm tripod is still the best way to ensure sharpness. With SR switched off, unless the instructions advise differently. Of course, reducing camera shake is fine, but for some subjects a high shutter speed will still be needed to reduce subjects movement. This might be wildlife, notorious for not staying still, and even leaves and branches swaying in the wind for landscape shots.
How fast a shutter speed? On full frame, using a 50mm standard lens then received wisdom will tell us 1/50s is the slowest we should go. For critical sharpness then 1/100s might be better. This will depend on how steady our hands are, and the effects of any shake reduction. Even without SR, if I need 1/8s I'll try anyway if I have to, but take several shots. One might just be OK. With APS-C it is suggested that the crop factor is used to re-calculate that minimum shutter speed. So 1/50s would become 1/75s, and for critical sharpness I might make it 1/150s. Again, notwithstanding and shake reduction.
A few pictures now where shutter speed has been a major factor.
1/250s to arrest swaying of foliage
1/250s so everyone is sharp
1/800s to freeze bird movement
1/8000s to freeze the jump
1/15s and shake reduction keeps things sharp
0.7s to get the required movement in the water
1/4000s water sports
1/2000s wildlife
1/6s Deliberate subject movement